Regular temperament: Difference between revisions

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A '''regular temperament''' ('''RT''') is an abstract [[tuning system]] that ~~looks the same no matter which pitch you start from (or consider the~~ [[~~tonic~~]]~~). In other words, unlimited free modulation is possible: any [[interval]] can be stacked as many times as you like. A regular temperament is [[generate]]d by~~ a set of ~~generating~~ intervals~~, usually one of which is considered the [[period]],~~ and ~~any note which is part of the regular temperament can be reached by stacking whole numbers of these generating intervals~~ above a defined root note. ~~Regular~~ temperaments theoretically have an infinite number of ~~notes~~, and besides [[equal temperament]]s, regular temperaments usually<ref group="note">This is true if there exist two generators such that size in [[cent]]s of one generator divided by that of the other is an {{W|irrational number}}. This is not true for tunings where every generator is a whole number of steps of some [[edo]] or other [[equal-step tuning]].</ref> have an infinite number of ~~notes~~ in between ''any two other ~~notes~~''.

A '''regular temperament''' ('''RT''') is an abstract [[tuning system]] that consists of all intervals reachable by arbitrarily [[stacking]] a set of intervals and their inverses above a defined root note. As such, regular temperaments theoretically have an infinite number of intervals, and besides [[equal temperament]]s, regular temperaments usually<ref group="note">This is true if there exist two generators such that size in [[cent]]s of one generator divided by that of the other is an {{W|irrational number}}. This is not true for tunings where every generator is a whole number of steps of some [[edo]] or other [[equal-step tuning]].</ref> have an infinite number of intervals in between ''any two other intervals''.

In addition ~~to unlimited modulation~~, regular ~~temperaments are~~ by definition ~~thought of as being approximations~~ of some system of pure or target intervals, very often a [[just intonation~~]] (JI) [[~~subgroup]]~~. Each~~ abstract interval is interpreted as a tempered, or detuned, version of a set of target intervals~~. A temperament only qualifies as a regular temperament if this interpretation works in a perfectly consistent way: The~~ product of two tempered intervals must always be the tempered version of the product of the JI intervals~~; for~~ example, if the ratios [[3/2]] and [[5/4]] are in the target interval set, then ~3/2 × ~5/4 = ~[[15/8]] must always be true. ("~" denotes tempered.) In any temperament, each target interval is mapped to a unique tempered interval, though a tempered interval can represent multiple target intervals.

In addition, a regular temperament is by definition an approximation of some system of pure or target intervals, very often a [[just intonation subgroup]], in a perfectly consistent way: each abstract interval is interpreted as a tempered, or detuned, version of a set of target intervals, and the product of two tempered intervals must always be the tempered version of the product of the JI intervals. For example, if the ratios [[3/2]] and [[5/4]] are in the target interval set, then ~3/2 × ~5/4 = ~[[15/8]] must always be true. ("~" denotes tempered.) In any temperament, each target interval is mapped to a unique tempered interval, though a tempered interval can represent multiple target intervals.

One particularly simple kind of regular temperaments is equal temperaments, which represent all intervals by multiples of a single step size. JI itself can be considered a [[trivial temperament]] where no tempering is happening: No commas are tempered out, and all of them are preserved as small pitch differences~~. Another example of a trivial temperament is [[single-pitch tuning]], where there are ''no'' generating intervals, and only a single pitch is available~~. In between JI and equal temperaments lies the cornucopia of temperaments discussed in [[Paul Erlich]]'s seminal work, [[:File:MiddlePath2015.pdf|''A Middle Path Between Just Intonation and the Equal Temperaments'']].

As a result, a regular temperament looks the same no matter which pitch you start from (or consider the [[tonic]]). In other words, unlimited free modulation is possible: any interval can be stacked as many times as you like.

One particularly simple kind of regular temperaments is equal temperaments, which represent all intervals by multiples of a single step size. JI itself can be considered a [[trivial temperament]] where no tempering is happening: No commas are tempered out, and all of them are preserved as small pitch differences. In between JI and equal temperaments lies the cornucopia of temperaments discussed in [[Paul Erlich]]'s seminal work, [[:File:MiddlePath2015.pdf|''A Middle Path Between Just Intonation and the Equal Temperaments'']].

== Examples ==

[[Meantone]] is the most historically significant regular temperament. It is generated by the [[octave]] and a tempered (detuned) version of the [[3/2|perfect fifth]], with the octave usually being considered the period, and every interval in meantone can be expressed as an integer number of octaves plus an integer number of fifths. In meantone, a [[major second]] is equal to two perfect fifths minus an octave, and a [[major third]] is four perfect fifths minus two octaves.

[[Meantone]] is the most historically significant regular temperament. It is generated by the [[octave]] and a tempered (detuned) version of the [[3/2|perfect fifth]], with the octave usually being considered the [[period]], and every interval in meantone can be expressed as an integer number of octaves plus an integer number of fifths. In meantone, a [[major second]] is equal to two perfect fifths minus an octave, and a [[major third]] is four perfect fifths minus two octaves.

The octave in meantone represents the just ratio [[2/1]], the perfect fifth 3/2, and the major third 5/4. Certain intervals are tempered to the [[unison]], or [[tempering out|tempered out]]; in a regular temperament, these intervals are known as [[comma]]s. In meantone, since stacking up four perfect fifths, down two octaves, and down a major third reaches the unison, we get that {{nowrap|(3/2)<sup>4</sup> / (2/1)<sup>2</sup> / (5/4) {{=}} [[81/80]]}} is tempered out, and thus 81/80 is a comma of meantone. Any two just intervals separated by a comma of a temperament, for example [[9/8]] and [[10/9]] in meantone, are mapped to the same tempered interval in the temperament, in this case a major second.

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 {{Wikipedia}}
-A '''regular temperament''' ('''RT''') is an abstract [[tuning system]] that looks the same no matter which pitch you start from (or consider the [[tonic]]). In other words, unlimited free modulation is possible: any [[interval]] can be stacked as many times as you like. A regular temperament is [[generate]]d by a set of generating intervals, usually one of which is considered the [[period]], and any note which is part of the regular temperament can be reached by stacking whole numbers of these generating intervals above a defined root note. Regular temperaments theoretically have an infinite number of notes, and besides [[equal temperament]]s, regular temperaments usually<ref group="note">This is true if there exist two generators such that size in [[cent]]s of one generator divided by that of the other is an {{W|irrational number}}. This is not true for tunings where every generator is a whole number of steps of some [[edo]] or other [[equal-step tuning]].</ref> have an infinite number of notes in between ''any two other notes''.
+A '''regular temperament''' ('''RT''') is an abstract [[tuning system]] that consists of all intervals reachable by arbitrarily [[stacking]] a set of intervals and their inverses above a defined root note. As such, regular temperaments theoretically have an infinite number of intervals, and besides [[equal temperament]]s, regular temperaments usually<ref group="note">This is true if there exist two generators such that size in [[cent]]s of one generator divided by that of the other is an {{W|irrational number}}. This is not true for tunings where every generator is a whole number of steps of some [[edo]] or other [[equal-step tuning]].</ref> have an infinite number of intervals in between ''any two other intervals''.
-In addition to unlimited modulation, regular temperaments are by definition thought of as being approximations of some system of pure or target intervals, very often a [[just intonation]] (JI) [[subgroup]]. Each abstract interval is interpreted as a tempered, or detuned, version of a set of target intervals. A temperament only qualifies as a regular temperament if this interpretation works in a perfectly consistent way: The product of two tempered intervals must always be the tempered version of the product of the JI intervals; for example, if the ratios [[3/2]] and [[5/4]] are in the target interval set, then ~3/2 × ~5/4 = ~[[15/8]] must always be true. ("~" denotes tempered.) In any temperament, each target interval is mapped to a unique tempered interval, though a tempered interval can represent multiple target intervals.
+In addition, a regular temperament is by definition an approximation of some system of pure or target intervals, very often a [[just intonation subgroup]], in a perfectly consistent way: each abstract interval is interpreted as a tempered, or detuned, version of a set of target intervals, and the product of two tempered intervals must always be the tempered version of the product of the JI intervals. For example, if the ratios [[3/2]] and [[5/4]] are in the target interval set, then ~3/2 × ~5/4 = ~[[15/8]] must always be true. ("~" denotes tempered.) In any temperament, each target interval is mapped to a unique tempered interval, though a tempered interval can represent multiple target intervals.
-One particularly simple kind of regular temperaments is equal temperaments, which represent all intervals by multiples of a single step size. JI itself can be considered a [[trivial temperament]] where no tempering is happening: No commas are tempered out, and all of them are preserved as small pitch differences. Another example of a trivial temperament is [[single-pitch tuning]], where there are ''no'' generating intervals, and only a single pitch is available. In between JI and equal temperaments lies the cornucopia of temperaments discussed in [[Paul Erlich]]'s seminal work, [[:File:MiddlePath2015.pdf|''A Middle Path Between Just Intonation and the Equal Temperaments'']].
+As a result, a regular temperament looks the same no matter which pitch you start from (or consider the [[tonic]]). In other words, unlimited free modulation is possible: any interval can be stacked as many times as you like.
+One particularly simple kind of regular temperaments is equal temperaments, which represent all intervals by multiples of a single step size. JI itself can be considered a [[trivial temperament]] where no tempering is happening: No commas are tempered out, and all of them are preserved as small pitch differences. In between JI and equal temperaments lies the cornucopia of temperaments discussed in [[Paul Erlich]]'s seminal work, [[:File:MiddlePath2015.pdf|''A Middle Path Between Just Intonation and the Equal Temperaments'']].
 == Examples ==
-[[Meantone]] is the most historically significant regular temperament. It is generated by the [[octave]] and a tempered (detuned) version of the [[3/2|perfect fifth]], with the octave usually being considered the period, and every interval in meantone can be expressed as an integer number of octaves plus an integer number of fifths. In meantone, a [[major second]] is equal to two perfect fifths minus an octave, and a [[major third]] is four perfect fifths minus two octaves.
+[[Meantone]] is the most historically significant regular temperament. It is generated by the [[octave]] and a tempered (detuned) version of the [[3/2|perfect fifth]], with the octave usually being considered the [[period]], and every interval in meantone can be expressed as an integer number of octaves plus an integer number of fifths. In meantone, a [[major second]] is equal to two perfect fifths minus an octave, and a [[major third]] is four perfect fifths minus two octaves.
 The octave in meantone represents the just ratio [[2/1]], the perfect fifth 3/2, and the major third 5/4. Certain intervals are tempered to the [[unison]], or [[tempering out|tempered out]]; in a regular temperament, these intervals are known as [[comma]]s. In meantone, since stacking up four perfect fifths, down two octaves, and down a major third reaches the unison, we get that {{nowrap|(3/2)<sup>4</sup> / (2/1)<sup>2</sup> / (5/4) {{=}} [[81/80]]}} is tempered out, and thus 81/80 is a comma of meantone. Any two just intervals separated by a comma of a temperament, for example [[9/8]] and [[10/9]] in meantone, are mapped to the same tempered interval in the temperament, in this case a major second.